1. Patterns, Equations, and Graphs Section 1-9

2. Goals Goal To use tables, equations, and graphs to describe relationships. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary Solution of an equation Inductive reasoning

4. The coordinate plane is formed by the intersection of two perpendicular number lines called axes. The point of intersection, called the origin, is at 0 on each number line. The horizontal number line is called the x-axis, and the vertical number line is called the y-axis. Review: Graphing in the Coordinate Plane

5. Points on the coordinate plane are described using ordered pairs. An ordered pair consists of an x-coordinate and a y-coordinate and is written (x, y). Points are often named by a capital letter. The x-coordinate tells how many units to move left or right from the origin. The y-coordinate tells how many units to move up or down. Reading Math Graphing in the Coordinate Plane

6. Graph each point. A. T(–4, 4) Start at the origin. Move 4 units left and 4 units up. B. U(0, –5) Start at the origin. Move 5 units down. T(–4, 4) U(0, –5) C. V (–2, –3) Start at the origin. Move 2 units left and 3 units down. V(–2, −3) Example: Graphing in the Coordinate Plane

7. Graph each point. A. R(2, – 3) B. S(0, 2) Start at the origin. Move 2 units right and 3 units down. Start at the origin. Move 2 units up. C. T( – 2, 6) Start at the origin. Move 2 units left and 6 units up. R(2, –3) S(0,2) T(–2,6) Your Turn:

8. The axes divide the coordinate plane into four quadrants. Points that lie on an axis are not in any quadrant. Graphing in the Coordinate Plane

9. Name the quadrant in which each point lies. A. E Quadrant ll B. F no quadrant (y-axis) C. G Quadrant l D. H Quadrant lll E F H G x y Example: Locating Points

10. Name the quadrant in which each point lies. A. T no quadrant (y-axis) B. U Quadrant l C. V Quadrant lll D. W Quadrant ll T W V U x y Your Turn:

11. The Rectangular Coordinate System SUMMARY: The Rectangular Coordinate System Composed of two real number lines – one horizontal (the x-axis) and one vertical (the y-axis). The x- and y-axes intersect at the origin. Also called the Cartesian plane or xy-plane. Points in the rectangular coordinate system are denoted (x, y) and are called the coordinates of the point. We call the x the x-coordinate and the y the y-coordinate. If both x and y are positive, the point lies in quadrant I; if x is negative, but y is positive, the point lies in quadrant II; if x is negative and y is negative, the point lies in quadrant III; if x is positive and y is negative, the point lies in quadrant IV. Points on the x-axis have a y-coordinate of 0; points on the y-axis have an x-coordinate of 0.

12. Equation in Two Variables An equation in two variables, x and y, is a statement in which the algebraic expressions involving x and y are equal. The expressions are called sides of the equation. Any values of the variables that make the equation a true statement are said to be solutions of the equation. x + y = 15x 2 – 2y 2 = 4y = 1 + 4x x + y = 15 The ordered pair (5, 10) is a solution of the equation. 5 + 10 = 15 15 = 15

13. Solutions to Equations 2x + y = 5 2(2) + (1) = 5 4 + 1 = 5 5 = 5 Example: Determine if the following ordered pairs satisfy the equation 2x + y = 5. a.) (2, 1)b.) (3, – 4) (2, 1) is a solution. True 2x + y = 5 2(3) + (– 4) = 5 6 + (– 4) = 5 2 = 5 (3, – 4) is not a solution. False

14. An equation that contains two variables can be used as a rule to generate ordered pairs. When you substitute a value for x, you generate a value for y. The value substituted for x is called the input, and the value generated for y is called the output. y = 10x + 5 OutputInput Equation in Two Variables

15. Table of Values Use the equation y = 6x + 5 to complete the table and list the ordered pairs that are solutions to the equation. xy(x, y) – 2 0 2 y = 6x + 5 x = – 2 y = 6(– 2) + 5 y = – 12 + 5 y = – 7 (– 2, – 7) – 7 y = 6x + 5 x = 0 y = 6(0) + 5 y = 0 + 5 y = 5 5 (0, 5) y = 6x + 5 x = 2 y = 6(2) + 5 y = 12 + 5 y = 17 17 (2, 17)

16. An engraver charges a setup fee of $10 plus $2 for every word engraved. Write a rule for the engraver ’ s fee. Write ordered pairs for the engraver ’ s fee when there are 5, 10, 15, and 20 words engraved. Let y represent the engraver’s fee and x represent the number of words engraved. Engraver ’ s fee is$10plus $2for each word y = 10 + 2 · x y = 10 + 2x Example: Application

17. The engraver’s fee is determined by the number of words in the engraving. So the number of words is the input and the engraver’s fee is the output. Writing Math

18. Number of Words Engraved RuleCharges Ordered Pair x (input)y = 10 + 2xy (output)(x, y) y = 10 + 2(5) 5 20 (5, 20) y = 10 + 2(10) 1030(10, 30) y = 10 + 2(15) 15 40 (15, 40) y = 10 + 2(20) 20 50 (20, 50) Example: Solution

19. What if…? The caricature artist increased his fees. He now charges a $10 set up fee plus $20 for each person in the picture. Write a rule for the artist’s new fee. Find the artist’s fee when there are 1, 2, 3 and 4 people in the picture. y = 10 + 20x Let y represent the artist’s fee and x represent the number of people in the picture. Artist ’ s fee is$10plus $20for each person y = 10 + 20 · x Your Turn:

20. Number of People in Picture RuleCharges Ordered Pair x (input)y = 10 + 20xy (output)(x, y) y = 10 + 20(1) 1 30 (1, 30) y = 10 + 20(2) 250(2, 50) y = 10 + 20(3) 3 70 (3, 70) y = 10 + 20(4) 490 (4, 90) Solution:

21. When you graph ordered pairs generated by a function, they may create a pattern. Graphing Ordered Pairs

22. Generate ordered pairs for the function using the given values for x. Graph the ordered pairs and describe the pattern. y = 2x + 1; x = –2, –1, 0, 1, 2 –2–2 –1–1 0 1 2 2( – 2) + 1 = – 3 ( – 2, – 3) ( – 1, – 1) (0, 1) (1, 3) (2, 5) 2( – 1) + 1 = – 1 2(0) + 1 = 1 2(1) + 1 = 3 2(2) + 1 = 5 InputOutput Ordered Pair xy(x, y) The points form a line. Example: Graphing Ordered Pairs

23. –4–4 –2–2 0 2 4 – 2 – 4 = – 6 ( – 4, – 6) ( – 2, – 5) (0, – 4) (2, – 3) (4, – 2) – 1 – 4 = – 5 0 – 4 = – 4 1 – 4 = – 3 2 – 4 = – 2 InputOutput Ordered Pair xy(x, y) The points form a line. y = x – 4; x = –4, –2, 0, 2, 4 1212 Your Turn: Generate ordered pairs for the function using the given values for x. Graph the ordered pairs and describe the pattern.

24. Definition Inductive Reasoning – is the process of reaching a conclusion based on an observed pattern. –Can be used to predict values based on a pattern.

25. Inductive Reasoning Moves from specific observations to broader generalizations or predictions from a pattern. Steps: 1.Observing data. 2.Detect and recognizing patterns. 3.Make generalizations or predictions from those patterns. Observation Pattern Predict

26. Make a prediction about the next number based on the pattern. 2, 4, 12, 48, 240 Answer: 1440 Find a pattern: 2 4 12 48 240 ×2×2 The numbers are multiplied by 2, 3, 4, and 5. Prediction: The next number will be multiplied by 6. So, it will be (6)(240) or 1440. ×3×3×4×4×5×5 Example: Inductive Reasoning

27. Make a prediction about the next number based on the pattern. Answer: The next number will be Your Turn:

28. Joke Time What’s a fresh vegetable? One that insults a farmer. What did one knife say to the other? Look sharp! What did one strawberry say to the other? “Look at the jam you’ve gotten us into!”

29. Assignment 1.9 Exercises Pg. 72 – 74: 8 – 54 even